Problem: Evaluate the infinite series $\sum_{k = 1}^\infty \frac{k^2}{2^k}.$
Explanation: Let
\[S = \sum_{k = 1}^\infty \frac{k^2}{2^k} = \frac{1^2}{2} + \frac{2^2}{2^2} + \frac{3^2}{2^3} + \frac{4^2}{2^4} + \dotsb.\]Then
\[2S = 1 + \frac{2^2}{2} + \frac{3^2}{2^2} + \frac{4^2}{2^3} + \frac{5^2}{2^4} + \dotsb.\]Subtracting these equations, we get
\[S = 1 + \frac{3}{2} + \frac{5}{2^2} + \frac{7}{2^3} + \frac{9}{2^4} + \dotsb.\]Then
\[2S = 2 + 3 + \frac{5}{2} + \frac{7}{2^2} + \frac{9}{2^3} + \frac{11}{2^4} + \dotsb.\]Subtracting these equations, we get
\[S = 4 + \frac{2}{2} + \frac{2}{2^2} + \frac{2}{2^3} + \frac{2}{2^4} + \dotsb = 4 + \frac{1}{1 - 1/2} = \boxed{6}.\]